P9-2004-218 П. Г. Акишин, А. В. Бутенко, А. Д. Коваленко, В. А

P9-2004-218
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‘µµ¡Ð¥´¨¥ ¡Ñ¥¤¨´¥´´µ£µ ¨´¸É¨ÉÊÉ Ö¤¥·´ÒÌ ¨¸¸²¥¤µ¢ ´¨°. „Ê¡´ , 2004
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Akishin P. G. et al.
Mathematical Simulation of the Magnetic Field End Effects
in Superconducting Nuclotron-Type Dipole Magnet
P9-2004-218
Mathematical model of a superconducting window-frame Nuclotron-type dipole
was developed within the frames of volume integral equations. 3D distributions
of the magnetic ˇeld in the magnet aperture were calculated for different operating
currents. The inuence of the magnet ferromagnetic yoke end proˇle (Rogovskytype) on the ˇeld nonlinearity is analyzed.
The investigation has been performed at the Veksler and Baldin Laboratory of
of High Energies and Laboratory of Information Technologies, JINR.
Communication of the Joint Institute for Nuclear Research. Dubna, 2004
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1. Œ’…Œ’ˆ—…‘ŠŸ ”Œ“‹ˆ‚Š ‹…Œ›
¸¸³µÉ·¨³ § ¤ ÎÊ ´ ̵¦¤¥´¨Ö · ¸¶·¥¤¥²¥´¨Ö ³ £´¨É´µ£µ ¶µ²Ö, ¸µ§¤ ¢ ¥³µ£µ ¸É ͨµ´ ·´Ò³¨ ɵ± ³¨ ¢ ³ £´¨É´ÒÌ ¸¨¸É¥³ Ì ¸ ¨§µÉ·µ¶´Ò³ Ë¥··µ³ £´¨É´Ò³ ³ É¥·¨ ²µ³. ʸÉÓ B(a), H(a), M(a) ¥¸ÉÓ ¨´¤Ê±Í¨Ö, ´ ¶·Ö¦¥´´µ¸ÉÓ
¨ ´ ³ £´¨Î¥´´µ¸ÉÓ ³ £´¨É´µ£µ ¶µ²Ö ¢ ɵα¥ a. ‚ µÉ¸Êɸɢ¨¥ ¶µ¢¥·Ì´µ¸É´ÒÌ
1
ɵ±µ¢ ¨ ɵ±µ¢, ¶·µÉ¥± ÕÐ¨Ì ¶µ Ë¥··µ³ £´¥É¨±Ê, ¢¥²¨Î¨´Ò B, H ʤµ¢²¥É¢µ·ÖÕÉ Ê· ¢´¥´¨Ö³ Œ ±¸¢¥²² :
rota (H(a)) = JS (a),
diva (B(a)) = 0,
lim |B(a)| = 0,
|a|→∞
(1)
£¤¥ JS Å ¢¥±Éµ· µ¡Ñ¥³´µ° ¶²µÉ´µ¸É¨ ɵ± (µÉ²¨Î¥´ µÉ ´Ê²Ö ɵ²Ó±µ ¢ µ¡³µÉ± Ì ³ £´¨É ). ‚¥²¨Î¨´Ò B, H, M ¸¢Ö§ ´Ò ¸²¥¤ÊÕШ³¨ ´¥²¨´¥°´Ò³¨
¸µµÉ´µÏ¥´¨Ö³¨:
H(a) =
B(a)
,
µ (|B(a)|) µ0
M(a) =
B(a)
− H(a),
µ0
(2)
£¤¥ µ0 Å ¡¸µ²ÕÉ´ Ö ³ £´¨É´ Ö ¶·µ´¨Í ¥³µ¸ÉÓ ¢ ±Êʳ , µ(x) Å ³ £´¨É´ Ö
¶·µ´¨Í ¥³µ¸ÉÓ (¢´¥ ¦¥²¥§´µ£µ Ö·³ ɵ¦¤¥¸É¢¥´´µ · ¢´ Ö ¥¤¨´¨Í¥, ¢´ÊÉ·¨
¦¥²¥§´µ£µ Ö·³ Å ´¥²¨´¥°´ Ö ËÊ´±Í¨Ö, Ì · ±É¥·¨§ÊÕÐ Ö ¸¢Ö§Ó ³¥¦¤Ê H
¨ B ¤²Ö ¤ ´´µ£µ ɨ¶ Ë¥··µ³ £´¥É¨± ). £· ´¨Í¥ · §¤¥² ¸·¥¤ ¸ · §²¨Î´Ò³¨ ³ £´¨É´Ò³¨ Ì · ±É¥·¨¸É¨± ³¨ ¢Ò¶µ²´ÖÕÉ¸Ö Ê¸²µ¢¨Ö ´¥¶·¥·Ò¢´µ¸É¨
´µ·³ ²Ó´µ° ¨ É ´£¥´Í¨ ²Ó´µ° ¸µ¸É ¢²ÖÕÐ¨Ì ¢¥±Éµ·µ¢ H ¨ B ¸µµÉ¢¥É¸É¢¥´´µ:
(n, (B2 − B1 )) = 0,
[n × (H2 − H1 )] = 0,
(3)
£¤¥ n Å ¥¤¨´¨Î´Ò° ¢¥±Éµ· ´µ·³ ²¨ ± ¶µ¢¥·Ì´µ¸É¨ · §¤¥² ¸·¥¤.
“· ¢´¥´¨Ö (1)Ä(3) ´¥Ê¤µ¡´Ò ¤²Ö ´¥¶µ¸·¥¤¸É¢¥´´µ£µ ´ ̵¦¤¥´¨Ö ¢¥±Éµ·ËÊ´±Í¨° H(a), B(a). ‘ÊÐ¥¸É¢Ê¥É ¡µ²Óϵ¥ Ψ¸²µ ¶µ¸É ´µ¢µ±, ¢ÒÉ¥± ÕШÌ
¨§ Ê· ¢´¥´¨° Œ ±¸¢¥²² [5]. ‚ ¤ ´´µ° · ¡µÉ¥ ¨¸¶µ²Ó§Ê¥É¸Ö ¨´É¥£· ²Ó´ Ö
¶µ¸É ´µ¢± § ¤ Ψ ³ £´¨Éµ¸É ɨ±¨ [6]:
1
1
H(a) = HS (a) +
∇a
(4)
M(x), ∇a
dνx ,
4π
|x − a|
Ω
£¤¥ H ¨ M ʤµ¢²¥É¢µ·ÖÕÉ ´¥²¨´¥°´Ò³ ¸µµÉ´µÏ¥´¨Ö³ (2).
„²Ö ¤¨¸±·¥É¨§ ͨ¨ ´¥²¨´¥°´ÒÌ µ¡Ñ¥³´ÒÌ ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨° ¶·¥¤² £ ¥É¸Ö ¨¸¶µ²Ó§µ¢ ÉÓ ³¥Éµ¤ ±µ²²µ± ͨ¨. ‘²¥¤ÊÖ [7], · §µ¡Ó¥³ µ¡² ¸ÉÓ Ω,
§ ¶µ²´¥´´ÊÕ Ë¥··µ³ £´¥É¨±µ³, ´ ¶µ¤³´µ¦¥¸É¢ {Ωl }:
Ω=
N
Ωl .
l=1
Œ¥· ¶¥·¥¸¥Î¥´¨Ö Ωl c Ωm · ¢´ ´Ê²Õ ¶·¨ l = m. ‚ ± ¦¤µ³ Ωl ¢ ± Î¥¸É¢¥
ɵα¨ ´ ¡²Õ¤¥´¨Ö al ¢Ò¡¥·¥³ Í¥´É·µ¨¤ Ωl :
xdνx
Ωi
.
al = dνx
Ωl
2
·¨¡²¨§¨³ ´ ³ £´¨Î¥´´µ¸ÉÓ M(x) ¢ ± ¦¤µ³ Ô²¥³¥´É¥ Ωi ¶µ¸ÉµÖ´´Ò³ ¢¥±Éµ·µ³ Mi .
’µ£¤ ¤¨¸±·¥É¨§µ¢ ´´ Ö ¸¨¸É¥³ Ê· ¢´¥´¨° (4) ¨³¥¥É ¸²¥¤ÊÕШ° ¢¨¤:


N
∇a 
1

S
Hl = H (al ) +
dν
M
,
∇
,
(5)

j
a
x 
4π
|x − a|
j=1
Ωj
a=al

1
HS (al ) = rota 
4π
ΩS

J(x)

dνx |x − a|
.
(6)
a=al
‚¥±Éµ·Ò Hl ¨ Ml ʤµ¢²¥É¢µ·ÖÕÉ ´¥²¨´¥°´Ò³ ¸µµÉ´µÏ¥´¨Ö³ (2).
‚¢¥¤¥³ ¸²¥¤ÊÕШ¥ µ¡µ§´ Î¥´¨Ö. ʸÉÓ
B̂ = (B1 , B2 , ..., BN )T , M̂(B̂) = (M(B1 ), M(B2 ), ..., M(BN ))T ,
ĤS = (HS (a1 ), HS (a2 ), ..., HS (aN ))T .
¶·¥¤¥²¨³ ³ É·¨ÍÊ [A] ¸²¥¤ÊÕШ³ µ¡· §µ³:


[A11 ] ... [A1N ]
...
...  ,
[A] =  ...
[AN 1 ] ... [AN N ]
£¤¥ [Aij ] Å ³ É·¨Í · §³¥·´µ¸É¨ [3 × 3], É ± Ö, Îɵ ¤²Ö ²Õ¡µ£µ ¶µ¸ÉµÖ´´µ£µ
¢¥±Éµ· M ¸¶· ¢¥¤²¨¢µ ¸µµÉ´µÏ¥´¨¥


∇a 
1

M, ∇a
[Aij ] M =
.
dνx 

4π
|x − a|
Ωj
a=ai
“ΨÉÒ¢ Ö (2), ¸¨¸É¥³Ê (5) ³µ¦´µ § ¶¨¸ ÉÓ ¢ ¢¨¤¥
B̂ = µ0 ĤS + ([A] + [E]) M̂(B̂),
(7)
£¤¥ [E] Å ¥¤¨´¨Î´ Ö ³ É·¨Í · §³¥·´µ¸É¨ [3N × 3N ].
„²Ö ·¥Ï¥´¨Ö ¸¨¸É¥³Ò Ê· ¢´¥´¨° (7) ¶·¨³¥´Ö¥É¸Ö ¸²¥¤ÊÕШ° ¨É¥· ͨµ´´Ò° ¶·µÍ¥¸¸:
B̂k+1 = µ0 ĤS + ([A] + [E]) M̂(B̂k+1 ),
(8)
B̂0 = µ0 ĤS , k = 0, 1, 2, ...
3
(9)
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³ £´¨É .
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¶·µË¨²Ö µ£µ¢¸±µ£µ
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¶¥·¥¸Î¥É ¶µ²Ö ¤²Ö µ¤´µ£µ §´ Î¥´¨Ö
ɵ± ¢ µ¡³µÉ± Ì ¸µ¸É ¢¨²µ µ±µ²µ
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‚¢¨¤Ê ¸²µ¦´µ¸É¨ ¶µ¢¥¤¥´¨Ö ¶µ²Ö
´ ±· Õ ³ £´¨É ¤²Ö ¶µ²ÊÎ¥´¨Ö ·¥Ï¥´¨Ö ´¥²¨´¥°´µ° ¸¨¸É¥³Ò (7) ¢
Ôɵ³ ¸²ÊÎ ¥ ¶µÉ·¥¡µ¢ ²µ¸Ó µ±µ²µ
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2 ¨ 2,5 ¸³ ¶·¨ §´ Î¥´¨¨ ¶µ²Ö ¢ Í¥´É·¥ ³ £´¨É 2 ’² ¶·¨¢¥¤¥´Ò ¢ É ¡²¨Í¥.
5
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(¸¶²µÏ´ Ö ²¨´¨Ö) ¶·¨ §´ Î¥´¨ÖÌ Éµ±µ¢ ¢ µ¡³µÉ± Ì ¢µ§¡Ê¦¤¥´¨Ö: a, ¢ Å 1,2 ± (¶µ²¥
0,42 ’²); ¡, £ Å 6,1 ± (¶µ²¥ 2 ’²)
ƒ¥µ³¥É·¨Ö
ƒ ·³µ´¨±a
ɵ·Í¥¢µ°
Î ¸É¨
B1 (R = 2 ¸³/2,5 ¸³) B3 (R = 2 ¸³/2,5 ¸³)
B5 (R = 2 ¸³/2,5 ¸³)
¡ÒδҰ
¶µ²Õ¸
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